Optimal. Leaf size=173 \[ -\frac {3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+2);\frac {1}{6} (3 m+8);\cos ^2(c+d x)\right )}{b d (3 m+2) \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}-\frac {3 C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+5);\frac {1}{6} (3 m+11);\cos ^2(c+d x)\right )}{b d (3 m+5) \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {20, 3010, 2748, 2643} \[ -\frac {3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+2);\frac {1}{6} (3 m+8);\cos ^2(c+d x)\right )}{b d (3 m+2) \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}-\frac {3 C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+5);\frac {1}{6} (3 m+11);\cos ^2(c+d x)\right )}{b d (3 m+5) \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 20
Rule 2643
Rule 2748
Rule 3010
Rubi steps
\begin {align*} \int \frac {\cos ^m(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{4/3}} \, dx &=\frac {\sqrt [3]{\cos (c+d x)} \int \cos ^{-\frac {4}{3}+m}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b \sqrt [3]{b \cos (c+d x)}}\\ &=\frac {\sqrt [3]{\cos (c+d x)} \int \cos ^{-\frac {1}{3}+m}(c+d x) (B+C \cos (c+d x)) \, dx}{b \sqrt [3]{b \cos (c+d x)}}\\ &=\frac {\left (B \sqrt [3]{\cos (c+d x)}\right ) \int \cos ^{-\frac {1}{3}+m}(c+d x) \, dx}{b \sqrt [3]{b \cos (c+d x)}}+\frac {\left (C \sqrt [3]{\cos (c+d x)}\right ) \int \cos ^{\frac {2}{3}+m}(c+d x) \, dx}{b \sqrt [3]{b \cos (c+d x)}}\\ &=-\frac {3 B \cos ^{1+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (2+3 m);\frac {1}{6} (8+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (2+3 m) \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}}-\frac {3 C \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5+3 m);\frac {1}{6} (11+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{b d (5+3 m) \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.38, size = 140, normalized size = 0.81 \[ -\frac {3 \sqrt {\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+2}(c+d x) \left (B (3 m+5) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+2);\frac {1}{6} (3 m+8);\cos ^2(c+d x)\right )+C (3 m+2) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+5);\frac {1}{6} (3 m+11);\cos ^2(c+d x)\right )\right )}{d (3 m+2) (3 m+5) (b \cos (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right ) + B\right )} \left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \cos \left (d x + c\right )^{m}}{b^{2} \cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{m}\left (d x +c \right )\right ) \left (B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{m}}{\left (b \cos \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^m\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (B + C \cos {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )} \cos ^{m}{\left (c + d x \right )}}{\left (b \cos {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________